Constructions, loci and geometric conditions · Foundation · 作图、轨迹与几何条件 · 基础
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| locus/ˈləʊkəs/ | 轨迹 | guǐ jì |
Where can both conditions hold?
- A router must be equally far from two rooms and within reach of a power point. Each condition creates a different set of possible positions.
- This lesson studies locus 轨迹: The set of all points satisfying a stated geometric condition.
Choose the mathematical structure
- Points equally distant from A and B lie on the perpendicular bisector of AB. Points at fixed distance r from C lie on a circle. Points equally distant from two intersecting lines lie on their angle bisectors.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines locus? · 以下哪项描述正确定义了轨迹？
The set of all points satisfying a stated geometric condition. · 满足特定几何条件的所有点的集合。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For A=(0,0) and B=(6,0), the perpendicular bisector is x=3. Points also 5 units from A satisfy x²+y²=25. Substituting x=3 gives y²=16, so (3,4) and (3,-4) satisfy both conditions.
Constructions, loci and geometric conditions · 作图、轨迹与几何条件
Points equally distant from A and B lie on the perpendicular bisector of AB · 到 A 和 B 等距的点位于 AB 的垂直平分线上
Explain why (3,4) belongs to both loci. · 解释为何 (3,4) 同时属于两个轨迹。
Find the midpoint x-coordinate between (0,0) and (6,0). · 求 (0,0) 与 (6,0) 之间中点的 x 坐标。
Midpoint x-coordinate=(0+6)/2=3. · 中点 x 坐标=(0+6)/2=3。
Test a tempting shortcut
- The perpendicular bisector concerns distance to two points; the angle bisector concerns distance to two lines. A sketch is not a ruler-and-compass construction: preserve arcs as evidence of the method.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Points equally distant from two points always lie on their angle bisector. This claim is false. Explain which definition or assumption it violates.
Find the positive y-coordinate on x=3 and x²+y²=25. · 求在 x=3 和 x²+y²=25 上的正 y 坐标。
At x=3, y²=25-9=16. The positive y-coordinate is 4. · 当 x=3 时，y²=25-9=16。该正 y 坐标为 4。
Points equally distant from two points always lie on their angle bisector. · 到两点等距的点始终位于它们的角平分线上。
The perpendicular bisector concerns distance to two points; the angle bisector concerns distance to two lines. A sketch is not a ruler-and-compass construction: preserve arcs as evidence of the method. · 垂直平分线涉及的是到两点的距离，而角平分线涉及的是到两条直线的距离。草图并非尺规作图：需保留圆弧作为方法的证据。
Interpret a new situation
- Translate each condition into a locus before finding intersections. For a region closer to A than B, choose the correct side of the perpendicular bisector and show whether a boundary is allowed.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the distance from (3,4) to (0,0). · 求从 (3,4) 到 (0,0) 的距离。
Distance=√(3²+4²)=5. · 距离=√(3²+4²)=5。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- 4MA1 · Foundation · 4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The set of all points satisfying a stated geometric condition. Choose the relationship, show the method, check its assumptions and interpret the result.